Abstract
Let \(\mathcal {V}\) be the variety of square-increasing idempotent semirings. Its members can be viewed as semilattice-ordered monoids satisfying \(x\le x^{2}\). We show that the universal theory of \(\mathcal {V}\) is decidable. In order to prove this result, we investigate the class \(\mathcal {Q}\) whose members are ordered-monoid subreducts of members from \(\mathcal {V}\). In particular, we prove that finitely generated members from \(\mathcal {Q}\) are well-partially-ordered and residually finite.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berstel, J., Reutenauer, C.: Rational Series and Their Languages, volume 12 of EATCS Monographs. Springer, Berlin (1988)
Blok, W.J., van Alten, C.J.: The finite embeddability property for residuated lattices, pocrims and BCK-algebras. Algebra Universalis 48(3), 253–271 (2002)
Blok, W.J., van Alten, C.J.: On the finite embeddability property for residuated ordered groupoids. Trans. Am. Math. Soc. 357(10), 4141–4157 (2005)
Bloom, S.L.: Varieties of ordered algebras. J. Comput. Syst. Sci. 13(2), 200–212 (1976)
Brian, A., Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order. Cambridge University Press, Cambridge (1990)
de Luca, A., Varricchio, S.: Finiteness and Regularity in Semigroups and Formal Languages. Springer, Berlin (1999)
Ehrenfeucht, A., Haussler, D., Rozenberg, G.: On regularity of context-free languages. Theor. Comput. Sci. 27(3), 311–332 (1983)
Evans, T.: Some connections between residual finiteness, finite embeddability and the word problem. J. Lond. Math. Soc. 1(1), 399–403 (1969)
Galatos, N., Jipsen, P.: Residuated frames with applications to decidability. Trans. Am. Math. Soc. 365(3), 1219–1249 (2013)
Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics, vol. 151 of Studies in Logic and the Foundations of Mathematics. Elsevier, Amsterdam (2007)
Higman, G.: Ordering by divisibility in abstract algebras. Proc. Lond. Math. Soc. 2, 326–336 (1952)
Ono, H.: Completions of algebras and completeness of modal and substructural logics. In: Balbiani, P., Suzuki, N., Wolter, F., Zakharyaschev, M. (eds.) Advances in Modal Logic, vol. 4, pp. 335–353. King’s College Publications, Cambridge (2003)
Pigozzi, D.: Partially ordered varieties and quasivarieties. http://orion.math.iastate.edu/dpigozzi/ (2003). Revised notes of lectures on joint work with Katarzyna Palasinska given at the CAUL, Lisbon in September of 2003, and at the Universidad Catolica, Santiago in November of 2003
Pin, J.-É.: Syntactic semigroups. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. 1, pp. 679–746. Springer, Berlin (1997). [chapter 10]
Richter-Gebert, J., Sturmfels, B., Theobald, T.: First steps in tropical geometry. In: Idempotent Mathematics and Mathematical Physics. Proceedings of the international workshop, Vienna, Austria, February 3–10, 2003, pp. 289–317. Providence, RI: American Mathematical Society (AMS) (2005)
Rosenstein, Joseph G.: Linear Orderings. Academic Press, New York (1982)
Schützenberger, M-P.: Une théorie algébrique du codage. Séminaire Dubreil. Algèbre et Théorie des Nombres, 9:1–24, 1955–1956
Acknowledgements
The author wishes to thank Petr Savický and anonymous referees for helpful comments and remarks. The work of the author was partly supported by the grant GAP202/11/1632 of the Czech Science Foundation and partly by the long-term strategic development financing of the Institute of Computer Science (RVO:67985807).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Jean-Eric Pin.
Rights and permissions
About this article
Cite this article
Horčík, R. On square-increasing ordered monoids and idempotent semirings. Semigroup Forum 94, 297–313 (2017). https://doi.org/10.1007/s00233-017-9853-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00233-017-9853-x